large eddy simulation

Monday, June 12, 2006

Large Eddy Simulation/Immersed boundaries/Cartyesian grid papers

Selected Publications

F. Tessicini
Ph.D. Thesis: Wall model for immersed boundaries/cartesian grid methods.

CTR: Wall modeling for large-eddy simulation using an immersed boundary

F. Tessicini, L. Temmerman and M. A. Leschziner
Approximate near-wall treatments based on zonal hybrid RANS-LES methods for LES at high Reynolds numbers
Accepted for publication. IJHFF

S. Leonardi, P. Orlandi, F. Tessicini, R. Antonia.
Title : DNS and LES of turbulent flows over rough surfaces
Accepted for publication. AIAA Journal

F. Tessicini, M.A. Leschziner
Approximate near-wall treatments based on zonal RANS-LES hybrid methods at high Reynolds numbers
Symposium on Hybrid RANS-LES Methods Rica City Hotel, Stockholm, 14-15 July, 2005
F. Tessicini, N. LI, M. Leschziner
Hybrid LES/RANS modelling of separated flow around a three-dimensional hill
Proceedings of DLES6, Ercoftac 2005

F.Tessicini, L. Temmerman, M. Leschziner
“Approximate near-wall treatments based on zonal and hybrid RANS-LES methods for LES at high Reynolds numbers”
Proceedings of ETMM6 Engineering Turbulence Modelling and Measurements, 2005

S. Leonardi, F. Tessicini, P. Orlandi, R.A. Antonia
“Direct and large eddy simulation of a turbulent channel flow with roughness on one wall”
Proceedings of the Fifteenth Australasian Fluid Mechanics Conference 2004

S. Leonardi, F. Tessicini, P. Orlandi, R.A. Antonia
“Large eddy simulation of a turbulent channel flow with square bars on one wall”
ETC10, Euromech European Turbulence Conference, 2004

F. Tessicini, R. Verzicco, P. Orlandi
Book Chapter:
"Nozzle Geometry Effects in the Near Field of a Round Jet"
Manipulation and Control of Jets in Crossflow
Series: CISM International Centre for Mechanical Sciences, Number 439
Karagozian, Ann R.; Cortelezzi, Luca; Soldati, Alfredo (Eds.)
2003, 309 p.,ISBN: 3-211-00753-9

F. Tessicini, R. Verzicco,G. Iaccarino,M Fatica
"Wall modeling for large eddy simulation using an immersed boundary method"
CTR Center for Turbulence Research, Annual research briefs, 2002

F. Tessicini, R. Verzicco, P. Orlandi, S. Leonardi
"DNS and LES of complex flows using an immersed boundary method
WCCM Proc. of the WCCM V, ISBN 3-9501554-0-6, 2002

F. Tessicini, R. Verzicco, P. Orlandi, S. Leonardi
"DNS and LES of Wakes Behind Cylinders with and without Undulations"
EUROMECH Colloquium No 433 Dynamics of Trailing Vortices
March,20-22 2002

F. Tessicini, R. Verzicco, P. Orlandi
"Initial Condition Effects in the Near Field of a Round Jet"
AIMETA 15th AIMETA Congress in
Taormina. AIMETA, Kluwer, September-01 2001.

F. Tessicini,P. Orlandi
"Numerical simulation in the Terni valley by ARPS" QNET-CFD
Proceedings 1st Workshop, June 10 2001

R. Verzicco,P.G. Esposito, F.Tessicini, P. Orlandi, S. Leonardi
"Numerical simulation of flow around a naval propeller at low Reynolds number"
ATI Annual meeting
September,15-20 2000.

F. Tessicini, R. Verzicco, P. Orlandi
"Numerical Simulation of Wakes Interacting with Bluff Bodies" Applied
Mathematics for Industrial Flow Second International conference
October,12 2000.

F. Tessicini, R. Verzicco, P. Orlandi, S. Leonardi
"Numerical Simulation of Wakes Interacting with Bluff Bodies" In-Vento
2000 6st National Conference Wind Engineering
June,18 2000.

The Zonal Two-layer Approach

The Zonal Two-layer Approach

The objective of the zonal strategy is to provide the LES region with the wall-shear stress, extracted from a separate modelling process applied to the near-wall layer. The wall-shear stress can be determined from an algebraic law-of-the-wall model or from differential equations solved on a near-wall-layer grid refined in the wall-normal direction - an approach referred to as two-layer wall modelling. The method was originally proposed by Balaras and Benocci and tested by Tessicini et al. At solid boundaries, the LES equations are solved up to a near-wall node which is located, typically, at $y^+=50$ . From this node to the wall, a refined mesh is embedded into the main flow,and the following simplified turbulent boundary-layer equations are solved:

$\frac{\partial{\rho\tilde{U}_i}}{\partial{t}}+ \frac{\partial{\rho\tilde{U}_i\tilde{U}_j}}{\partial{x_j}}+ \frac{d\tilde{P}}{dx_i} = \frac{\partial}{\partial{y}}[(\mu+\mu_t)\frac{\partial{\tilde{U}_i}}{\partial{y}}]\quad i=1,3$

where y denotes the direction normal to the wall and i identify the wall-parallel directions (1 and 3).

The eddy viscosity $\mu_t$ is obtained from a mixing-length model with near-wall damping, as done by Wang and Moin : $\frac{\mu_t}{\mu} = \kappa{y}_{w}^+(1-e^{-y_w^+/A})^2$ The boundary conditions for the turbulent boundary layer equations are given by the unsteady outer-layer solution at the first grid node outside the wall layer and the no-slip condition at y=0.

References:

Balaras E. and Benocci C. (1994) In: Applications of Direct and Large Eddy Simulation, AGARD. pp. 2-1-2-6.

Cabot W. and Moin P. (2000) Flow, Turbulence and Combustion, 63:269-291

Tessicini F., Temmerman L. and Leschziner M.A. (2005) In: 6th Engineering Turbulence Modelling and Measurements (ETMM6)

Thursday, June 08, 2006

Introduction

Large eddy simulation (LES) is a popular technique for simulating turbulent flows. An implication of Kolmogorov's (1941) theory of self similarity is that the large eddies of the flow are dependant on the geometry while the smaller scales more universal. This feature allows one to explicitly solve for the large eddies in a calculation and implicitly account for the small eddies by using a subgrid-scale model (SGS model).

Mathematically, one may think of separating the velocity field into a resolved and sub-grid part. The resolved part of the field represent the "large" eddies, while the subgrid part of the velocity represent the "small scales" whose effect on the resolved field is included through the subgrid-scale model. Formally, one may think of filtering as the convolution of a function with a filtering kernel $G$ :

$\bar{u}_i(\vec{x}) = \int G(\vec{x}-\vec{\xi}) u(\vec{x})d\vec{\xi},$

resulting in

$u_i = \bar{u}_i + u'_i,$

where $\bar{u}_i$ is the resolvable scale part and $u'_i$ is the subgrid-scale part. However, most practical (and commercial) implementations of LES use the grid itself as the filter (the box filter) and perform no explicit filtering. More information about the theory and application of filters is found in the LES filters article.

This page is mainly focused on LES of incompressible flows. For compressible flows, see Favre averaged Navier-Stokes equations.

The filtered equations are developed from the incompressible Navier-Stokes equations of motion:

$\frac{\partial{u_i}}{\partial t} + u_j \frac{\partial u_i}{\partial x_j} = -\frac{1}{\rho} \frac{\partial p}{\partial x_i} + \frac{\partial}{\partial x_j}\left(\nu \frac{\partial u_i}{\partial x_j}\right).$

Substituting in the decomposition $u_i = \bar{u}_i + u'_i$ and $u_i = \bar{p} + p'$ and then filtering the resulting equation gives the equations of motion for the resolved field:

$\frac{\partial{\bar{u}_i}}{\partial t} + \bar{u}_j \frac{\partial \bar{u}_i}{\partial x_j} = -\frac{1}{\rho} \frac{\partial \bar{p}}{\partial x_i} + \frac{\partial}{\partial x_j}\left(\nu \frac{\partial \bar{u}_i}{\partial x_j}\right) + \frac {1}{\rho}\frac{\partial \tau_{ij}}{\partial x_j}.$

We have assumed that the filtering operation and the differentiation operation commute, which is not generally the case. It is thought that the errors associated with this assumption are usually small, though filters that commute with differentiation have been developed ("ref?"). The extra term $\frac{\partial \tau_{ij}}{\partial x_j}$ arises from the non-linear advection terms, due to the fact that

$\overline{ u_j \frac{\partial u_i}{\partial x_j} } \ne \bar{u}_j \frac{\partial \bar{u}_i}{\partial x_j}$

and hence

$\tau_{ij} = \bar{u}_i \bar{u}_j - \overline{u_i u_j}$

Similar equations can be derived for the subgrid-scale field (i.e. the residual field).

Subgrid-scale turbulence models usually employ the Boussinesq hypothesis, and seek to calculate (the deviatoric part of) the SGS stress using:

$\tau _{ij} - \frac{1}{3}\tau _{kk} \delta _{ij} = - 2\mu_t \bar S_{ij}$

where $\bar S_{ij}$ is the rate-of-strain tensor for the resolved scale defined by

$\bar S_{ij} = \frac{1}{2}\left( {\frac{{\partial \bar u_i }}{{\partial x_j }} + \frac{{\partial \bar u_j }}{{\partial x_i }}} \right)$

and $\nu_t$ is the subgrid-scale turbulent viscosity. Substituting into the filtered Navier-Stokes equations, we then have

$\frac{\partial{\bar{u}_i}}{\partial t} + \bar{u}_j \frac{\partial \bar{u}_i}{\partial x_j} = -\frac{1}{\rho} \frac{\partial \bar{p}}{\partial x_i} + \frac{\partial}{\partial x_j}\left([\nu+\nu_t]\frac{\partial\bar{u}_i}{\partial x_j}\right),$

where we have used the incompressibility constraint to simplify the equation and the pressure is now modified to include the trace term $\tau _{kk} \delta _{ij}/3$ .

[edit]

Subgrid-scale models

Smagorinsky model (Smagorinsky, 1963)
Algebraic Dynamic model (Germano, et. al., 1991)
Localized Dynamic model (Kim & Menon, 1993)
WALE (Wall-Adapting Local Eddy-viscosity) model (Nicoud and Ducros, 1999)
RNG-LES model

[edit]

References

J. Smagorinsky. General circulation experiments with the primitive equations, i. the basic experiment. Monthly Weather Review, 91: 99-164, 1963.
M. Germano, U. Piomelli, P. Moin, and W. H. Cabot. A dynamic sub-grid scale eddy viscosity model. Physics of Fluids, A(3): 1760-1765, 1991.
W. Kim and S. Menon. A new dynamic one-equation subgrid-scale model for large eddy simulation. In 33rd Aerospace Sciences Meeting and Exhibit, Reno, NV, 1995.
F. Nicoud and F. Ducros. Subgrid-scale modelling based on the square of the velocity gradient tensor. Flow, Turbulence and Combustion, 62: 183-200, 1999.

large eddy simulation

Monday, June 12, 2006